We study theoretically the irreversible quenching of an excitation migrating on a polymer by small quencher molecules. Using a simplified superposition approximation, we solve the hierarchical reaction-diffusion equations and obtain the quenching rate and survival probability of the excitation. Depending upon the excitation mobility as well as upon the chain length, the quenching rate shows various behaviors, which is absent in the small molecular quenching reaction. In the diffusion-controlled limit, the quenching rate is found to increase as the excitation mobility increases and converge, for sufficiently large excitation mobility, to a constant value proportional to the radius of gyration of the polymer. This means that fast migration of single excitation, assisted by the dynamics of the polymer, enhance the reaction rate as much as that of the reaction of a static polymer where all the monomers are reactive.